Plastic Design by Moment Balancing

Two theorems are important in the plastic analysis of rigid-frame structures. These are usually called the upper bound theorem and the lower-bound theorem. According to the upper-bound theorem, the load corresponding to an assumed mechanism of collapse will always be greater than, or at best equal to, the true ultimate load. According to the lower-bound theorem, the load corresponding to a given moment diagram for which the moment nowhere exceeds the plastic-moment capacity of the frame will always be less than, or at most equal to, the true ultimate load. If one constructs a moment diagram for a structure—any moment diagram, so long as it satisfies statics—and proportions the structure to this moment diagram so that there are enough plastic hinges to form a mechanism, the upperbound and lower-bound theorems are satisfied simultaneously. This conclusion is the basis for a scheme proposed by Horne1 that is particularly suited for purposes of design. It can be used to construct a distribution of moments about which the structure can then be developed in such a way as to result in a sufficient number of plastic hinges to produce a mechanism. The procedure is very efficient in application to frames of several stories, and offers the designer liberty he does not enjoy in design based on elastic methods of analysis. The method is also useful for preliminary sizing of frames which are to be designed elastically.